CAPÍTULO II: EL PRECEDENTE CONSTITUCIONAL VINCULANTE EN EL
2. EL PRECEDENTE VINCULANTE:
2.7. CONCEPTOS RELACIONADOS AL PRECEDENTE VINCULANTE:
We utilize the theory of the grand canonical ensemble to treat stellar gases in the LTE (local thermodynamic equilibrium) condition. A system of a stellar gas is defined by a temperature T=l/p, a volume V and number of atoms of each element Np The total number of atoms Na is a sum of Nj over all elements. The corresponding number densities are
Chapter II The Equation of State with Nonideal Effects
defined by n^ = Na/V and n^ = N,/V.
First of all, we define a system which consists only of the bound electrons and free electrons. To establish such a system in a stellar gas, the radiation and nuclei have to be separated from it. This is done in the following ways. (1) The radiation is treated as a photon gas, independent of the others. (2) The nuclei are treated as classical particles since their interaction with bound electrons is contributed to the bound states of electrons. (3) The interactions of the nuclei with free electrons are also included in the energy states of the free electrons, and the interactions among neutral particles are described by the hard sphere m odel. Therefore, we only need to consider the distribution of electrons in both bound and free states.
2.4.1.2 Electron partition function
We start with an electron partition function (denoted by EPF) for the system. Since the electron distribution involves the number distribution of electrons in different states, it is better to resort to the theory of the grand canonical ensemble which includes the number operator in the partition function. In statistical physics, the grand canonical partition function is defined by
ZG = Trexp(X,A-pA) .
Since the partition function for the electrons is concerned, the number operator A, the energy operator È and the degeneracy parameter X should be those of the electrons. In a representation of the eigenstates with the electron energies and numbers {Ej, Ni), in which exp(XA-pA) is diagonal, Zq is obtained by summing exp(%A-pA) over all eigenstates and performing its product over all subsystems. The subsystem can be one atomic system, or one element, or one phase space cell which can hold one electron.
2.4.1.3 EPF for bound states
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Chapter II The Equation of State with Nonideal Effects
We consider the electrons bound to the nuclei. The EPF in an atomic configuration of element i with m bound electrons and eigenvalue E^^^j for state j is given by
Zimj = < m, Eimjl exp(A,^-pA) I m, Eimj>
= exp(Xm-PEimj) .
Here it is noted that the configuration energy E ^ j is the total energy of all bound electrons in the configuration. The perturbation of bound electron energy levels should also be included in Ei^^j by an energy shift AEn. For instance, the configuration energy of a hydrogen-like atom in the state ln> is written as
Fimj = “ Z ~ 2 ^Fn . 2n^
The first term is the configuration energy of the unperturbed atom while the second term is due to the perturbation of the bound electron. For a configuration with multiple electrons, the total energy is shifted by the perturbation of all bound electrons in the atom.
The statistical effect of the finite size of atoms is incorporated into the EPF by multiplying by an occupation probability Wi^j» expressed in terms of = -loge(Wjjnj)- Thereby, summing Zj^ij over all eigenstates {j) and all possible numbers of bound electrons {m) gives the EPF in one atom of the element i, i.e.
Zb(i) — Z Zjmj Wimj m J
Z Z exp(km-PEimj-4>imj) • m J
The EPF for the bound states of all atoms of all elements is obtained by performing a product of Zb(i) over all atoms of all elements, i.e.
Chapter II The Equation of State with Nonideal Effects
with the number of atoms Nj of element i. Its logarithm has a form
loge Zb = Z Ni loge [ Z Z exp(Xm-PEinij-<|)imj) ] ,
1 m j
with a sum over all elements.
2.4.1.4 EPF for free states
We also need the EPF for the free states. Since there are two eigenstates for a free electron in a unit cell of phase space, i.e. there can be one electron or none, we then have for the EPF per unit cell with energy E
Zi = <m,melexp(X,N-pH)lm,me> with m=0,l
=1 + exp(A,-PE).
The EPF for all free states is given by finding a product of Zj over the whole free phase space, that is
Z f= n z i . <free>
Its logarithmic form is
logeZf = £ loge [ 1 + exp(A,-Pe) ] .
<free>
The sum here has to be converted into an integration over the whole phase space available to the free electron
Z =
J
47cp2 dp ,<free> (2ti) 0
Chapter II The Equation of State with Nonideai Effects
with a statistical weight g^=2 for two states of electron spin and the electron momentum p.
The energy of a free electron £ includes its kinetic energy and its potential energy including the interaction with other charged particles. If we represent this interaction by a statistically averaged energy decreased by AEf, then the energy of the free electron is equal to
£ = ^ - A E f.
2.4.1.5 Atomic configuration probability
Now we have got the EPF, Zy and Zf, in both bound and free states. According to statistical physics, the total number of electrons in the system is defined by
I Ni Zi = |-loge{ZfZb) i dAi S m 2 exp(Xm-PE^.-<^.^.) ^ F i/2 ( X + P A E f ) + 2 N i --- 2 ? exp(\m-PE^.-1..^.) = Nf + 2 N i 2 i ; P. . , i m j
where F1/2% is the Fermi-Dirac integral. The left hand side is from the conservation of
electron number while the right hand side represents a statistical distribution of electrons in
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LTE conditions. So it can be called the conservation equation for the electron number. Thisequation can be solved for the electron degeneracy parameter X when given N^^ Zj, p, V, Fimj» AEf and
In the conservation equation, the first term on the right hand side is the number of free electrons, and the second one is the number of electrons bound with nuclei. The second term thus defines a probability for a configuration of element i with m bound electrons and
Chapter II The Equation of Slate with Nonideal Effects
t 1
i-'.i
eigenvalue Ejj^j to be
exp(Xm-PE imj imj.)
The Saha and Boltzmann equations are found to be special cases of this formula with the inclusion of an occupation probability.